Question Video: Simplifying a Complex Number Expression Using Conjugates Mathematics

Video Transcript

Simplify negative three plus two 𝑖 times three plus three 𝑖 over four plus 𝑖 times four plus four 𝑖.

There are four complex numbers in this question. We have the product of two complex numbers up here and the product of two complex numbers down here. Then we recall that the fraction line means divide. And so what we’re going to need to do is perform both sets of multiplication and then divide the results.

Let’s begin by considering the pair of complex numbers on the numerator of our fraction. We have negative three plus two 𝑖 times three plus three 𝑖. And we multiply a pair of complex numbers just as we multiply a pair of binomials. We’ll multiply the first term in each expression. Negative three times three is negative nine. Then we multiply the outer terms. Negative three times three 𝑖 is negative nine 𝑖. We multiply the inner terms. Two 𝑖 times three is six 𝑖. And we multiply the last term in each expression. Two 𝑖 times three 𝑖 is six 𝑖 squared.

We collect like terms, and we see that our expression simplifies to negative nine minus three 𝑖 plus six 𝑖 squared. We’re not quite finished though. We know that 𝑖 squared is equal to negative one. And so we can rewrite the term six 𝑖 squared as six times negative one. And so we have negative nine minus three 𝑖 plus six times negative one. And since six times negative one is negative six, this simplifies to negative 15 minus three 𝑖.

Let’s repeat this process with the denominator. Multiplying four by four gives us 16. We multiply the outer terms to get 16𝑖 and the inner terms to get four 𝑖. Then multiplying the last terms gives us four 𝑖 squared. 16𝑖 plus four 𝑖 is 20𝑖. And then we’re going to rewrite four 𝑖 squared as four times negative one, which is negative four. And so the product of four plus 𝑖 and four plus four 𝑖 is 12 plus 20𝑖.

Now that we’ve worked out the product of our pairs of complex numbers, we know we need to divide them. We’re going to be dividing negative 15 minus three 𝑖 by 12 plus 20𝑖. And of course, to divide a pair of complex numbers, we write the quotient as a fraction and then we multiply both the numerator and the denominator of that fraction by the conjugate of the denominator.

Now, for a complex number 𝑧 of the form 𝑎 plus 𝑏𝑖, its conjugate is 𝑎 minus 𝑏𝑖. Essentially, we simply change the sign of the imaginary part. This means the conjugate of our denominator is 12 minus 20𝑖. And so we’re going to need to perform a very similar process as to the one when we multiplied our pairs of complex numbers. We’re going to multiply negative 15 minus three 𝑖 by 12 minus 20𝑖 and 12 plus 20𝑖 by 12 minus 20𝑖.

Let’s begin with the pair of complex numbers on the numerator of our new fraction. Distributing as we did before, and we get negative 180 plus 300𝑖 minus 36𝑖 plus 60𝑖 squared. We can rewrite this as negative 180 plus 264𝑖 plus 60 times negative one and then further rewrite that last term as negative 60. And so the product of negative 15 minus three 𝑖 and 12 minus 20𝑖 is negative 240 plus 264𝑖.

Let’s repeat this process with the denominator. Distributing and we get 144 minus 240𝑖 plus 240𝑖 minus 400𝑖 squared. Well, negative 240𝑖 plus 240𝑖 is zero. So this becomes 144 minus 400𝑖 squared, which is 144 minus 400 times negative one. Negative 400 times negative one is positive 400. And so the product of this complex number and its conjugate simplifies to 544. And this means when we simplify this fraction, we get to negative 240 plus 264𝑖 over 544.

We might notice that we can simplify this a little further. So let’s clear some space. We’re going to separate the fraction up so it looks a little bit more like the general form of a complex number. And we get negative 240 over 544 plus 264 over 544 𝑖. Our last job is just to simplify both of these fractions. And when we do, we find that the result of multiplying our pairs of complex numbers and then dividing their products is negative 15 over 34 plus 33 over 68 𝑖.